Let’s assume that is continuous and positive on the interval . Then the definite integral represents the area of the region bounded by the graph of and the x-axis, from x = a to x = b. First, we partition the interval into n subintervals, each of width such that Then we can form a trapezoid for each subinterval and the area of the ith trapezoid = . This implies that the sum of the areas of the n trapezoids is Area =
Let S and S‘ be the following subsets of the plane: and
a) Show that S’ is an equivalence relation on the real line and that .
Transitivity- If then and thus z-y is the difference of two integers which implies that z-y is itself an integer.
To show that we note that which
b) Show that given any collection of equivalence relations on a set A, their intersection is an equivalence relation in A.
Proof: Let be a nonempty class of equivalence relations and let
Reflexivity- If then .
Transitivity– If then the intersection is an equivalence relation on A.